Pennsylvania State University employees are protesting a new wellness program that requires them to provide detailed health information or pay penalties that can total $1,200 a year, in an unusually public backlash against an increasingly common employer practice.

An online petition calling for the program to be canceled, launched by Penn State art-history professor Brian Curran, has gathered more than 2,000 signatures. Associate professor of political science Matthew Woessner has called publicly, in a letter on a...